Violation of the first law of black hole thermodynamics in f(T) gravity
aa r X i v : . [ h e p - t h ] N ov USTC-ICTS-11-08
Violation of the first law of black hole thermodynamics in f ( T ) gravity Rong-Xin Miao a , Miao Li b , and Yan-Gang Miao ca Interdisciplinary Center for Theoretical Study,University of Science and Technology of China,Hefei, Anhui 230026, People’s Republic of China. ∗ b Kavli Institute for Theoretical Physics,Key Laboratory of Frontiers in Theoretical Physics,Institute of Theoretical Physics, Chinese Academy of Sciences,Beijing 100190, People’s Republic of China. † and c School of Physics, Nankai University,Tianjin 300071, People’s Republic of China. ‡ Abstract
We prove that, in general, the first law of black hole thermodynamics, δQ = T δS , is violatedin f ( T ) gravity. As a result, it is possible that there exists entropy production, which impliesthat the black hole thermodynamics can be in non-equilibrium even in the static spacetime. Thisfeature is very different from that of f ( R ) or that of other higher derivative gravity theories. Wefind that the violation of first law results from the lack of local Lorentz invariance in f ( T ) gravity.By investigating two examples, we note that f ′′ (0) should be negative in order to avoid the nakedsingularities and superluminal motion of light. When f ′′ ( T ) is small, the entropy of black holes in f ( T ) gravity is approximatively equal to f ′ ( T )4 A . ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION f ( T ) gravity as a new modified gravity theory has recently attracted much attention [1–27]. It was first investigated by Ferraro and Fiorini [1, 2] in the Born-Infeld style which canlead to regular cosmological spacetimes without Big Bang singularity. Then, it was proposedby Bengochea, Ferraro and Linder [3, 4] to explain the current accelerated expansion ofuniverse. Similar to f ( R ) gravity, it is a generalization of the teleparallel gravity ( T G )[28–30] which was originally developed by Einstein in an attempt of unifying gravity andelectromagnetism. Let us make a brief review of
T G . The basic variables in
T G are tetradfields e aµ , where a is index of the internal space running over 0 , , , µ is the spacetimeindex running from 0 to 3. The tetrad fields are related with the spacetime metric by g µν = e aµ η ab e bν , η ab = e aµ e bν g µν = diag( − , , , . (1)In T G , the Weitzenbock connection Γ λµν = e λa ∂ ν e aµ (2)rather than the Levi-Civita connection is used to define the covariant derivative, and as aresult there is no curvature but only torsion T λµν = Γ λνµ − Γ λµν = e aλ ( ∂ µ e aν − ∂ ν e aµ ) . (3)The torsion scalar is defined by T = 12 S µνρ T µνρ = 14 T µνρ T µνρ + 12 T µνρ T ρνµ − T σµσ T ρρµ , (4)with the so-called dual torsion S µνρ = 12 ( T µνρ + T νµρ − T ρµν ) + g µρ T σνσ − g νρ T σµσ . (5)There are several virtues in T G . For example, in contrast to Einstein gravity, a covariantstress tensor of gravitation can naturally be defined in the gauge context of
T G [31].As a main advantage compared with f ( R ) gravity, the equations of motion of f ( T ) gravityare second-order instead of fourth-order. However, the local Lorentz invariance is violatedin f ( T ) gravity [18] and consequently more degrees of freedom appear. Recently, we haveinvestigated the Hamiltonian formulation of f ( T ) gravity and have found that three extra2egrees of freedom emerge [27]. In general, there are D − f ( T )gravity in D dimensions, and this implies that the extra degrees of freedom might correspondto one massive vector field. For the detailed explanation, see our recent work [27].In this paper, we investigate the black hole thermodynamics in f ( T ) gravity and findthat the first law, δQ = T δS , is violated. There is entropy production even in the staticspacetime and the black hole thermodynamics turns out to be non-equilibrium. By analyzingtwo examples in detail, we find that it is the violation of the local Lorentz invariance in f ( T )gravity that leads to the breakdown of the first law of black holes. Because of this violation,some degrees of freedom in f ( T ) gravity feel an effective metric different from the backgroundmetric. Consequently, they see a different horizon and Hawking temperature from that feltby matter fields with the local Lorentz invariance. Black holes in such a situation would notbe in equilibrium, thus it is not surprising that the first law is violated. In addition, fromthe two examples that will be analyzed, we also observe that f ′′ (0) should be negative inorder to avoid the naked singularities and super velocity of light.It should be stressed that, by “black hole” in f ( T ) gravity, we mean in the sense of theusual metric. Recently, some “black holes” in this sense were found in [32]. In general, theremay exist modes which can escape from the inside and make the horizon defined by themetric “non-black”. However, as shown in Appendix B, there indeed exist exact solutions of f ( T ) gravity which have the properties of the usual black hole. All the modes feel the samemetric and no modes can escape from the inside of the horizon. We focus on the “blackhole” in the metric in this paper.The paper is arranged as follows. In Sect. II, we give a brief review of the first law of f ( R ) gravity using the field equation method. In Sect. III, we establish the first law of f ( T )gravity. In Sect. IV, we search for the reasons for the violation of first law of f ( T ) gravityby investigating two examples. We conclude in Sect. V. II. FIRST LAW OF f ( R ) GRAVITY
The first law of black holes, δQ = T δS , is universal for gravity with the diffeomorphismLagrangian, L ( g µν , R µνρσ ), constructed from the metric g µν and Riemann tensor R µνρσ . Onecan derive the first law and entropy of black holes from various procedures, for instance,the Wald’s Noether charge method [33]. However, we shall use a different approach [34–37]3hich was originally developed to derive the gravity field equations from the thermodynamicpoint of view. In this paper we turn the logic around: we suppose the gravity field equationsand check if the thermodynamic relation δQ = T δS is satisfied. Though similar in someaspects, there are many differences between the Wald’s Noether charge approach and thefield equation approach. Here we just list three main differences. First, the Wald’s Noethercharge approach is based on the Lagrangian or the action of a theory, while the field equationapproach is based on the equations of motion. Second, the definitions of energy are differentin the two approaches. In the former, Wald uses the “canonical energy” E from which onecan derive δE = T δS + Ω H δJ , where Ω H is the angular velocity of the horizon and J is theangular momentum. In the latter, one defines the heat flux passing through the null surfaceas eq. (6), which does not contain the information of angular momentum. As we shall showbelow, using eq. (6), one can only derive δQ = T δS . Third, it is natural to use the fieldequation approach rather than the Wald’s approach to study the first law of black holesin f ( T ) gravity. The Wald’s approach is not designed for the teleparallel gravity. The keypoint of the field equation approach is the definition of the heat flux passing through thenull surface. According to ref. [18], we still have ∇ µ T µν = 0 in f ( T ) gravity, and thereforethe current T µν ξ µ remains conserved, i.e. ∇ µ ( T µν ξ ν ) = 0. Thus, it is very natural to useeq. (6) as the heat flux passing through the null surface in f ( T ) gravity.Now we give a brief review of the field equation approach. Let us take f ( R ) gravity asan example, and consider a heat flux δQ passing through an open patch on a null surface orblack hole horizon, dH = dAdλ , δQ = Z H T µν ξ µ k ν dAdλ, (6)where T µν is the matter stress-tensor, ξ µ is the Killing vector, H denotes the null surface, λ is the affine parameter, and k µ = dx µ dλ is the tangent vector to H . Substituting the f ( R )field equation f ′ ( R ) R µν − ∇ µ ∇ ν f ′ ( R ) + g µν (cid:18) (cid:3) f ′ ( R ) − f ( R ) (cid:19) = 8 πT µν (7)4nto eq. (6), we can derive δQ = 18 π Z H (cid:18) f ′ ( R ) R µν − ∇ µ ∇ ν f ′ ( R ) (cid:19) ξ µ k ν dAdλ = 18 π Z H (cid:18) f ′ ( R ) ∇ µ ∇ ν ξ µ − ξ µ ∇ µ ∇ ν f ′ ( R ) (cid:19) k ν dAdλ = 18 π Z H (cid:18) k ν ∇ µ (cid:0) f ′ ( R ) ∇ ν ξ µ (cid:1)(cid:19) dAdλ = 18 π Z H (cid:18) k ν l µ f ′ ( R ) ∇ ν ξ µ (cid:19) dAdλ = κ π (cid:16) f ′ ( R ) dA (cid:17)(cid:12)(cid:12)(cid:12) dλ = T δS. (8)In the above derivations, we have used Stokes’s Theorem and the following formulas: k µ ξ ν = 0 , k µ k µ = 0 , l µ l µ = 0 , k µ l µ = − , (9) R µν ξ µ = ∇ µ ∇ ν ξ µ , ξ µ ∇ µ R = 0 , (10) k µ l ν ∇ µ ξ ν = κ, T = κ π , dκdλ = 0 , (11)where κ is the surface gravity of the null surface H . From eq. (8), we can read out theentropy of black holes as S = f ′ ( R ) A , which is consistent with the Wald’s result.It should be stressed that in order to derive the first law, δQ = T δS , in eq. (8), wehave used the formula eq. (10) which is valid only for an exact Killing vector ξ . However,in general, there is no such an exact Killing vector in a dynamic spacetime. One can atmost obtain a Killing vector to the second order (in Riemann normal coordinates), ξ µ = − λk µ + O ( λ ), in our case [34]. Lack of an exact Killing vector implies that the spacetimemight be out of equilibrium and leads to the appearance of extra terms in eq. (8), whichcan be explained as contributions from entropy production in view of Jacobson’s idea [35].For simplicity, we focus on the cases with exact Killing vectors below. Note that the staticand stationary black holes always have an exact Killing vector, therefore our discussions areuniversal enough. The method mentioned above can easily be generalized to gravity withthe diffeomorphism Lagrangian, L ( g µν , R µνρσ ), constructed from the metric and Riemanntensor. Substituting the field equation P cdea R bcde − ∇ c ∇ d P acdb − Lg ab = 8 πT ab , P abcd = ∂L∂R abcd (12)5nto eq. (6), one can derive δQ = 18 π (cid:16) k a l b (cid:0) P abcd ∇ c ξ d − ξ d ∇ c P abcd (cid:1) dA (cid:17)(cid:12)(cid:12)(cid:12) dλ = T δS, (13)where S = κ ( P abcd ∇ c ξ d − ξ d ∇ c P abcd ) k a l b dA is equivalent to Wald entropy [38]. III. VIOLATION OF FIRST LAW OF f ( T ) GRAVITY
Now we use the field equation method introduced in Sec. II to investigate the first law ofblack hole thermodynamics in f ( T ) gravity. We find that the Clausius relation, dS = dQT , isviolated, which implies that even in a static spacetime the black hole of f ( T ) gravity is outof equilibrium and gives an intrinsic entropy production.Let us recall the equation of motion of f ( T ) gravity [18], H µν = f ′ ( T )( R µν − R g µν ) + 12 g µν [ f ( T ) − f ′ ( T ) T ] + f ′′ ( T ) S νµρ ∇ ρ T = 8 π Θ µν , (14) H [ µν ] = f ′′ ( T ) S [ νµ ] ρ ∇ ρ T = 0 , (15)where Θ µν is the matter stress-tensor. As the matter action is supposed to be invariantunder the local Lorentz transformation, Θ µν is symmetric and satisfies ∇ µ Θ µν = 0. Noticethat eq. (15) is just the antisymmetric part of eq. (14). According to ref. [18], eqs. (14) and(15) are not Lorentz invariant. This leads to an important fact that the solution of eqs. (14)and (15) is unique for every given Θ µν . Unlike Einstein gravity or T gravity, in general, onecannot get a new solution of eqs. (14) and (15) from the old one by performing local Lorentztransformations.The Hawking radiation is known to be independent of dynamics of gravity, which is apurely kinematic effect that is universal for Lorentz geometries containing an event horizon[39]. Thus, the Hawking temperature felt by matter (whose action has a local Lorentzinvariance) in f ( T ) gravity is the same as that in Einstein gravity, T = κ π . On the otherhand, the entropy of black holes is related to dynamics of gravity. Now let us begin to studythe first law and entropy of black holes in f ( T ) gravity, we still focus on the spacetimewith an exact Killing vector. By “Killing vector ξ µ ”, we mean in the sense of the usualmetric that it satisfies the equation, L ξ g µν = ξ α ∂ α g µν + ∂ µ ξ α g αν + ∂ ν ξ α g αµ = 0. Since onemetric corresponds to many different tetrad fields which are related with each other by local6orentz transformations, it is possible that the metric is static while the tetrad fields aretime dependentConsider a heat flux δQ passing through an open patch on a null surface or black holehorizon, we have δQ = Z H Θ µν ξ µ k ν dAdλ. (16)Substituting eq. (14) into the above equation, we get δQ = 18 π Z H k ν [ f ′ ( T ) R µν ξ µ + ξ µ S νµρ ∇ ρ f ′ ( T )] dAdλ = 18 π Z H k ν [ f ′ ( T ) ∇ µ ∇ ν ξ µ + ξ µ S νµρ ∇ ρ f ′ ( T )] dAdλ = 18 π Z H k ν [ ∇ µ ( f ′ ( T ) ∇ ν ξ µ ) − ( ∇ µ f ′ ( T )) ∇ ν ξ µ + ξ µ S νµρ ∇ ρ f ′ ( T )] dAdλ = κ π (cid:16) f ′ ( T ) dA (cid:17)(cid:12)(cid:12)(cid:12) dλ + 18 π Z H k ν ∇ µ f ′ ( T )( ξ ρ S ρνµ − ∇ ν ξ µ ) dAdλ. (17)Note that in the above derivations, we have used R µν ξ µ = ∇ µ ∇ ν ξ µ and ξ µ ∼ k µ on the nullsurface, and thus we have ξ µ k ν S µνρ = ξ ν k µ S µνρ . It should be mentioned that since ξ µ ∼ k µ on the null surface, only the symmetrical part of eq. (14) contributes to eq. (17), while theantisymmetric part eq. (15) does not contribute to eq. (17).The first term κ π ( f ′ ( T ) dA ) | dλ in the above equation is similar to the last line of eq. (8),therefore it can be explained as T δS . It is interesting that an extra term appears which ingeneral neither vanishes nor can be rewritten in the form R H k ν ∇ µ B [ νµ ] dAdλ for an arbitrary f ′ ( T ). We shall give the proof below.If the second term vanishes for an arbitrary f ′ ( T ), we then get k ν ξ ρ S ρνµ − k ν ∇ ν ξ µ = 0.However, due to the fact that k ν ∇ ν ξ µ is a Lorentz scalar but k ν ξ ρ S ρνµ is not, the twoterms can not be equal to each other. This contradiction shows that the second term ofeq. (17) is non-vanishing. Similarly, suppose that the second term can be rewritten as k ν ∇ µ B [ νµ ] for an arbitrary k µ (we can change the direction of k µ arbitrarily by choosinga different open patch of the null surface or choosing a different null surface), we have ∇ µ B [ νµ ] = ∇ µ f ′ ( T )( ξ ρ S ρνµ − ∇ ν ξ µ ). Note that ∇ ν ∇ µ B [ νµ ] = R µν B [ νµ ] = 0, we can obtain ∇ µ f ′ ( T ) ∇ ν ( ξ ρ S ρνµ − ∇ ν ξ µ ) = 0. For an arbitrary f ′ ( T ) we deduce ∇ ν ( ξ ρ S ρνµ − ∇ ν ξ µ ) = 0.Considering the fact that ∇ ν ∇ ν ξ µ is a local Lorentz scalar while ∇ ν ( ξ ρ S ρνµ ) is not, weconclude that the second term of eq. (17) would not take the form k ν ∇ µ B [ νµ ] . Notice thatwe do not use eq. (15) in the above derivations. One may guess that the last term of eq. (17)7anishes provided eq. (15) is used. However, it is not the case. There exist tetrad fields thatsatisfy eqs. (14) and (15) but still make the last line of eq. (17) non-vanishing. To end upthe proof, we give an example in Appendix A to show that the second term in the last lineof eq. (17) is indeed non-vanishing even provided eq. (15) is used.It should be mentioned that we have proved that, in general, the first law of black bolethermodynamics is violated for f ( T ) gravity. But there might exist some special casesin which the first law of f ( T ) black boles recovers. Note that for black holes with thesame metric g µν , we have many different choices of tetrad fields e aµ which are related witheach other by local Lorentz transformations. Those black holes have the same k ν ∇ ν ξ µ butdifferent k ν ξ ρ S ρνµ . Thus, for some special cases, the two terms might cancel each other andthe second term of the last line of eq. (17) vanishes. We give such an example in AppendixB in which the first law δQ = T δS recovers on the null surface.Similar to f ( R ) gravity [35], the second term of eq. (17) may be explained as contributionsfrom entropy production18 π Z H k ν ∇ µ f ′ ( T )( ξ ρ S ρνµ − ∇ ν ξ µ ) dAdλ = − T δS i , (18)which implies the black hole thermodynamics becomes non-equilibrium, δQ = T δS − T δS i .It should be stressed that there is one main difference between the entropy production of f ( R ) gravity and that of f ( T ) gravity. For f ( R ) gravity, when the Killing vector is exact (forexample, the static and stationary black holes), the entropy production vanishes. While for f ( T ) gravity, we find that there is entropy production even in a static spacetime. Since theentropy and entropy production should always be positive, there are very strict constraintsfor f ( T ) gravity, f ′ ( T ) > , f ′′ ( T ) k ν ∇ µ T ( ξ ρ S ρνµ − ∇ ν ξ µ ) ≤ . (19)The local Lorentz invariance has been examined by experiment in many sectors of thestandard model, including photons, electrons, protons and neutrons [40–42]. No violationof Lorentz symmetry has been identified so far in these sectors. M¨uller et al. performed anexperiment to test the local Lorentz symmetry in the gravitational sector and they founda small violation of local Lorentz invariance [43]. To be consistent with those experiments,the violation of the local Lorentz invariance in f ( T ) gravity should be very small. Note that f ′′ ( T ) can be used as a parameter to denote the violation of the local Lorentz invariance8ince it vanishes when f ( T ) gravity reduces to T G with local Lorentz invariance. So f ′′ ( T )is also expected to be very small and in that case the entropy production term eq. (18) canbe ignored. Thus, for a small f ′′ ( T ), the first law of black holes is satisfied approximativelyand the entropy is f ′ ( T )4 A .Finally, we observe that for the special case f ′ ( T ) = 1 the entropy production vanishesand the entropy reduces to that of Einstein gravity S = A , which is consistent with theequivalence between T G and Einstein gravity.
IV. REASON FOR VIOLATION OF FIRST LAW OF f ( T ) GRAVITY
In this section, we search for the reason for violation of first law of black holes in f ( T )gravity by investigating two concrete examples, Rindler space and Minkowski space. We findthat it is the violation of the local Lorentz invariance that leads to the breakdown of first lawof black holes in f ( T ) gravity. Although all the matter fields with the local Lorentz invariancesee the same horizon and Hawking temperature, some gravitational degrees of freedom in f ( T ) gravity feel a different background metric, horizon and Hawking temperature. Blackholes in such a situation cannot be in equilibrium [44–46] and consequently the first law inequilibrium is violated.For simplicity, we focus on f ( T ) gravity in 3 D below (The discussions below can be easilyextended to the 4 D case.). As the first example, let us consider the linear perturbation equa-tions of f ( T ) gravity with the background tetrad fields e aµ = diag( x, ,
1) and perturbations e aµ . The background spacetime is Rindler space with metric ds = − x dt + dx + dy ,while the perturbations of metric are h µν = e aµ η ab e bν + e aν η ab e bµ . For the sake of con-venience, we set f (0) = 0, which means that there is no cosmological constant term in theaction of f ( T ) gravity. Note that the background tetrad fields e aµ satisfy field equations of f ( T ) gravity in vacuum, and the background torsion scalar T = 0.We recall that the equation of motion of f ( T ) gravity is eq. (14), from which the linearperturbation equation can be derived in terms of T = 0, f ′ (0)2 [ ∇ ν ∇ ρ ¯ h ρµ + ∇ µ ∇ ρ ¯ h ρν − (cid:3) ¯ h µν − g µν ∇ ρ ∇ σ ¯ h ρσ ] + f ′′ (0) S ρνµ ∇ ρ T = 8 π Θ µν , (20)where ∇ µ is the covariant derivative defined by g µν , and (cid:3) = ∇ µ ∇ µ . T and Θ µν arethe perturbations of torsion scalar and stress tensor, respectively. ¯ h µν = h µν − h g µν and9 = h µν g µν . Similar to Einstein gravity, we can impose the Lorentz gauge ∇ µ ¯ h µν =0 tosimplify the above equation. The reason is that f ( T ) gravity is also invariant under thegeneral coordinate transformations, x µ → x µ + ζ µ , h µν → h µν + 2 ∇ ( µ ζ ν ) . For every given h µν , we can always find some suitable gauge parameters ζ µ to make h ′ µν = h µν + 2 ∇ ( µ ζ ν ) satisfy the Lorentz gauge ∇ µ ¯ h ′ µν =0 . In fact, we only need to solve the equation for ζ µ , −∇ µ ¯ h µν = (cid:3) ζ ν + R νµ ζ µ = (cid:3) ζ ν , where we have used R µν = 0 in Rindler space. It is clearthat solutions always exist for the above equation. Applying the Lorentz gauge ∇ µ ¯ h µν = 0,we can simplify eq. (20) as follows: − f ′ (0)2 (cid:3) ¯ h µν + f ′′ (0) S ρνµ ∇ ρ T = 8 π Θ µν . (21)Note that f ′′ (0) can be used to denote the violation of local Lorentz invariance, and thatwhen it vanishes the above perturbation equation recovers the local Lorentz invariance.Using background tetrad fields e aµ = diag( x, , S ρνµ (see eq. (5)). Thenon-zero results are given by S yyx = − x , (22) S xyy = 1 x . (23)From the antisymmetric part of eq. (21), S ρ [ νµ ] ∇ ρ T = 0, and eq. (22), we can derive ∂ y T = 0 . (24)As the simplest solution of the above equation, we require the perturbation e aµ be inde-pendent of coordinate y . Substituting eq. (23) into eq. (21), we find that most componentsof ¯ h µν obey the same equation as that in Einstein gravity, − f ′ (0)2 (cid:3) ¯ h µν = 8 π Θ µν , f ′ (0) = 1 , (25)except for ¯ h yy . For those fields that satisfy the same equation as that in Einstein gravity,they feel the same background metric (Rindler space in our case), therefore see the samehorizon and Hawking temperature as the matter fields.However, ¯ h yy satisfies a different equation in the form of − f ′ (0)2 (cid:3) φ + f ′′ (0) 1 x ∂ x T = 8 π Θ ′ , (26)10here φ stands for ¯ h yy , and Θ ′ = Θ yy . Note that ¯ h yy behaves like a scalar under the actionof (cid:3) in Rindler space, (cid:3) ¯ h yy = √− g ∂ ν ( √− gg νµ ∂ µ ¯ h yy ), thus we denote it by φ . For simplicity,we require that all e aµ vanish except for e (2) y = φ . Consequently, we have h yy = 2 φ ,¯ h yy = φ and T = 2 S aµν ∂ µ e aν − S avc T adc e dν = − x ∂ x φ . In view of ∂ y e aµ = 0, weobserve that this choice satisfies the Lorentz gauge ∇ µ ¯ h µν . Now, eq. (26) becomes − f ′ (0)2 (cid:3) φ − f ′′ (0) (cid:18) x ∂ x (cid:19) φ = 8 π Θ ′ . (27)It should be stressed that for our simple choice that all e aµ vanish except for e (2) y = φ , wehave ¯ h tt = x φ and ¯ h xx = − φ , which leads to two constraints for Θ tt and Θ xx from eq. (25).For simplicity, we require that Θ tt and Θ xx satisfy the constraints.Redefine φ = q x | ǫ + x | ¯ φ and Θ ′ = q x | ǫ + x | ¯Θ ′ , where ǫ = f ′′ (0) f ′ (0) , we can rewrite the aboveequation as − f ′ (0)2 [ ¯ (cid:3) − V ( x )] ¯ φ = 8 π ¯Θ ′ , (28)where V ( x ) = ǫ (3 ǫ +4 x )4 x ( ǫ + x ) , and ¯ (cid:3) is defined by the effective metric ¯ g µν which takes the form¯ g µν = − x x x + ǫ
00 0 1 . (29)As a result, the field ¯ φ feels an effective metric ¯ g µν different from that of Rindler space.If ǫ >
0, the horizon of this effective metric still lies at x = 0, and the Hawking temper-ature is T = 12 π N µ ∇ µ e ϕ = 12 πx √ ǫ + x , (30)where N µ = (0 , x √ ǫ + x ,
0) is a unit outward pointing vector normal to the horizon, ϕ = log( − ζ µ ζ µ ) is the Newton’s potential and ζ µ = (1 , ,
0) is a time-like Killing vector. Notethat the temperature T diverges at the horizon, and the worse is that there is a nakedsingularity at x = 0 in view of Ricci scalar ¯ R = ǫx . According to the cosmic censorshipconjecture, no naked singularities other than the Big Bang singularity exist in the universe.Therefore, in order to avoid the naked singularities and divergence of temperature, ǫ wouldnot be positive. 11or ǫ <
0, the position of the horizon turns to be x = √− ǫ , where ¯ R = ǫx , ¯ R µνρσ ¯ R µνρσ = ǫ x and ¯ R µν ¯ R µν = ǫ x have a good behavior and the singularity at x = 0 is hidden within thehorizon. The temperature T = 0 on the horizon x = √− ǫ can be read out from eq. (30),which is different from the temperature T = π felt by matter fields in Rindler space.Now let us summarize our results. At first, the scalar field ¯ φ in f ( T ) gravity feels aneffective metric eq. (29) different from that felt by matter fields, it therefore sees a differenthorizon and Hawking temperature. Black holes in such a situation would not be in theequilibrium state. Second, notice that the parameter ǫ = f ′′ (0) f ′ (0) is related to the violationof local Lorentz invariance. When ǫ vanishes, eq. (20) recovers the local Lorentz invarianceand the effective metric eq. (29) reduces to the metric of Rindler space. Furthermore, when f ′′ ( T ) = 0 and ǫ = 0, the entropy production terms (eq. (18)) vanish and the first law ofblack hole thermodynamics recovers. As a result, the breakdown of first law of black holesresults from the violation of local Lorentz invariance ( ǫ = 0). At last, ǫ should be negativein order to avoid the naked singularity.To end up this section, we briefly discuss the second example with background metric g µν = diag( − , ,
1) and tetrad fields e aµ = cosh( x ) sinh( x ) 0sinh( x ) cosh( x ) 00 0 1 . (31)Again, e aµ satisfy the field equations of f ( T ) gravity in vacuum when f (0) = 0. Note that T = 0 and the non-vanishing S ρµν are S yyt = S tyy = 1. After imposing the Lorentzgauge ∂ µ ¯ h µν = 0, we conclude that most of metric perturbations ¯ h µν obey the same equationeq. (25) as that in Einstein gravity expect for ¯ h yy which satisfies − f ′ (0)2 (cid:3) φ + f ′′ (0) ∂ t T = 8 π Θ yy , (32)where φ denotes ¯ h yy . Focusing on the case all perturbations of tetrad fields vanish expectfor e (2) y = φ , we have T = − ∂ t φ . Thus, the above equation becomes − f ′ (0)2 [ (cid:3) φ + ǫ ( ∂ t ) φ ] = 8 π Θ yy , (33)12rom which one can easily read out the effective metric ¯ g µν as follows:¯ g µν = − − ǫ . (34)From ds = ¯ g µν dx µ dx ν = 0, we get the speed of field φ , v = √ − ǫ . It is interesting that ifwe require that v does not exceed the speed of light, we get ǫ <
0, which is the same asthe condition in the first example given for getting rid of the naked singularity. It shouldbe mentioned that one can derive a similar condition in light of the recent work of Y. F.Cai et al. [23]. From eq. (28) of their paper [23], we note that both f ′′ ( T ) and ǫ = f ′′ (0) f ′ (0) ( f ′ (0) > c s doesnot exceed the speed of light. V. CONCLUSION
In this paper, we have shown that, in general, the first law of black hole thermodynamics δQ = T δS is violated in f ( T ) gravity, and only for some special cases can it be recovered.There is entropy production even in the static spacetime, and there are strict constraintsfor f ( T ) gravity in order to maintain the positivity of entropy and entropy production. Wefind that the violation of first law results from the lack of local Lorentz invariance in f ( T )gravity. Through investigating two concrete examples, we observe that the effective metricfelt by some degrees of freedom in f ( T ) gravity is different from the background metricfelt by matter fields because of the violation of local Lorentz invariance. The degrees offreedom therefore see a different horizon and Hawking temperature. Black holes in such asituation would not be in equilibrium, so it is the violation of local Lorentz invariance thatleads to the breakdown of the first law of black hole thermodynamics, δQ = T δS , in f ( T )gravity. To avoid the naked singularity and super velocity of light in the two examples,we get the condition ǫ = f ′′ (0) f ′ (0) <
0, where ǫ is a parameter which denotes the violation oflocal Lorentz invariance in f ( T ) gravity. To be consistent with experiments, ǫ and f ′′ ( T )should be small. In that case, the entropy production term is small compared with thefirst term in the last line of eq. (17), thus the first law of black hole thermodynamics canbe satisfied approximatively and the entropy of black holes in f ( T ) gravity equals f ′ ( T )4 A approximatively. 13 cknowledgements R-X.M. would like to thank T. Wang for useful discussions. M.L. and R-X.M. are sup-ported by the NSFC grants No.10535060, No.10975172 and No.10821504, and by the 973program grant No.2007CB815401 of the Ministry of Science and Technology of China. Y-G.M. is supported by the NSFC grant No.11175090.
Appendix A
We give the proof that the entropy production on the null surface is indeed non-vanishingby studying a specific example, Rindler space. The metric of Rindler space is ds = − x dt + dx + dy + dz , (35)where x ∈ [0 , ∞ ). We choose the following tetrad fields e aµ , x g ( x ) y ] sin [ g ( x ) y ] 00 − sin [ g ( x ) y ] cos [ g ( x ) y ] 00 0 0 1 (36)with an arbitrary function g ( x ) and the torsion scalar T = − g ( x ) x . One can check that theysatisfy the equations of motion eqs. (14) and (15) as long as the matter stress-tensor is givenby Θ = 116 πx (cid:2) x f ( T ) + 2 xgf ′ ( T ) + 4 g ( g − xg ′ ) f ′′ ( T ) (cid:3) , Θ = 18 π (cid:20) f ( T )2 + gf ′ ( T ) x (cid:21) , Θ = 18 π (cid:20) f x gf ′ ( T ) + 2( g − xg ′ ) f ′′ ( T ) x (cid:21) , Θ = 18 π (cid:20) f gf ′ ( T ) x + 2(1 + xg )( g − xg ′ ) f ′′ ( T ) x (cid:21) . (37)One can always choose suitable functions g ( x ) and f ( T ) to make Θ µν be regular and simul-taneously to keep the entropy production non-vanishing. For example, set g ( x ) = x e −| x | and f ( T ) = N P n =1 a n T n ( T = − x e −| x | ), where N is an arbitrary finite integer greater than 1,we find that the matter stress-tensor eq. (37) is regular in the whole space.14t should be stressed that there are many different choices of null surface in Rindlerspace. Without the loss of generality, we focus on the null surface x = e t below. Thecorresponding Killing vector and null vector on this null surface are ξ µ = (1 − cosh tx , sinh t, , k µ ∼ ( e t x , , , ξ µ ξ µ = ξ µ k µ = k µ k µ = 0, ξ µ ∼ k µ on this null surface. From eq. (18), we find that the entropy production on the null surface x = e t is proportional to − k ν ∇ µ f ′ ( T )( ξ ρ S ρνµ − ∇ ν ξ µ ) ∼ − f ′′ ( T ) e t ( xg ′ − g )[( x − cosh t ) g + 1] x , (38)which is non-vanishing generally. Appendix B
We give an exact solution of eqs. (14) and (15) which has the properties of the usualRindler space. All the modes feel the same Rindler space metric as that felt by matter fieldsand no modes can escape from inside of the horizon. Similar to Sect. IV, in order to get theeffective metric felt by the tetrad fields, let us investigate the linear perturbation equationsof f ( T ) gravity with the background tetrad fields e aµ = x cosh t sinh t x sinh t cosh t (39)and perturbations e aµ . The background spacetime is Rindler space with metric ds = − x dt + dx + dy + dz , while the perturbations of metric are h µν = e aµ η ab e bν + e aν η ab e bµ .Notice that we have T = 0 , S ρµν = 0 , (40)for the background tetrad fields eq. (39). Eq. (39) satisfies eqs. (14) and (15) provided thebackground matter stress-tensor is Θ µν = f (0)16 π δ µν . Using eq. (40), we can easily find thatthe linear perturbation of eq. (15) automatically vanishes. Thus, we only need to study thelinear perturbation of eq. (14). Following the approach of Sect. IV, we can derive − f ′ (0)2 (cid:3) ¯ h µν = 8 π Θ µν , (41)15hich is exactly the same as the linear perturbation equations of Einstein gravity if we set f ′ (0) = 1. Let us recall that ¯ h µν = h µν − h g µν and Θ µν is the linear perturbation ofthe matter stress-tensor. From eq. (41), it is clear that all the modes of tetrad fields feelthe same metric as the background spacetime, Rindler space. Thus, at least in the linearperturbation, all the modes feel the same horizon and null surfaces, and no modes can escapefrom inside of the horizon. From eqs. (17) and (40), it is interesting to note that the firstlaw δQ = T δS recovers on the null surface of Rindler space with tetrad fields eq. (39), andthe entropy on the null surface can be read out from eq. (17) as S = f ′ (0)4 A . [1] R. Ferraro and F. Fiorini, Phys. Rev. D , 084031 (2007) [arXiv:gr-qc/0610067].[2] R. Ferraro and F. Fiorini, Phys. Rev. D , 124019 (2008) [arXiv:0812.1981[gr-qc]].[3] G. R. Bengochea and R. Ferraro, Phys. Rev. D , 124019 (2009) [arXiv:0812.1205[astro-ph]].[4] E. V. Linder, Phys. Rev. D , 127301 (2010) [arXiv:1005.3039[astro-ph.CO]].[5] P. Wu and H. Yu, Phys. Lett. B , 415 (2010) [arXiv:1006.0674[gr-qc]].[6] R. Myrzakulov, Eur. Phys. J. C , 1752 (2011) [arXiv:1006.1120[gr-qc]].[7] P. Yu Tsyba, I. I. Kulnazarov, K. K. Yerzhanov and R. Myrzakulov, Int. J. Theor. Phys. ,1876 (2011) [arXiv:1008.0779[astro-ph.CO]].[8] P. Wu and H. Yu, Eur. Phys. J. C , 1552 (2011) [arXiv:1008.3669[gr-qc]].[9] K. Bamba, C. Q. Geng and C. C. Lee, [arXiv:1008.4036[astro-ph.CO]].[10] K. Bamba, C. Q. Geng and C. C. Lee, J. Cosmo. Astropart. Phys. , 021 (2010)[arXiv:1005.4574].[11] R. Myrzakulov, arXiv:1008.4486[astro-ph.CO].[12] P. Wu and H. Yu, Phys. Lett. B , 176 (2010) [arXiv:1007.2348[astro-ph.CO]].[13] K. Karami and A. Abdolmaleki, arXiv:1009.2459[gr-qc].[14] S. -H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, Phys. Rev. D , 023508 (2011)[arXiv:1008.1250[astro-ph.CO]].[15] J. B. Dent, S. Dutta and E. N. Saridakis, J. Cosmo. Astropart. Phys. , 009 (2011).[16] H. Wei, X. P. Ma and H. Y. Qi, Phys. Lett. B , 74 (2011) [arXiv:1106.0102[gr-qc]].[17] T. P. Sotiriou, B. Li and J. D. Barrow, Phys. Rev. D , 104030 (2011) [arXiv:1012.4039[gr-qc]].
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